Solve: [tex]4^{3 x}=4^2[/tex]
Answer (1)
3 2
Explanation
Understanding the Problem We are given the equation 4 3 x = 4 2 . Our goal is to solve for x . Since the bases are the same, we can equate the exponents.
Equating Exponents Equating the exponents, we have 3 x = 2 .
Solving for x To solve for x , we divide both sides of the equation by 3: x = 3 2
Final Answer Therefore, the solution to the equation 4 3 x = 4 2 is x = 3 2 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest is calculated using exponential functions. If you invest a principal amount P at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . Solving for t or r in such equations involves using logarithms, which are closely related to exponential functions. Understanding how to solve exponential equations is crucial for making informed financial decisions.
